Theorem unimax 2532

Description: Any member of a class is the largest of those members that it includes.

Assertion

Ref	Expression
unimax	|- (A e. B -> U.{x e. B | x (_ A} = A)

Distinct variable groups:   x,A   x,B

Proof of Theorem unimax

Step	Hyp	Ref	Expression
1		ssid 2080	. . 3 |- A (_ A
2		sseq1 2082	. . . 4 |- (x = A -> (x (_ A <-> A (_ A))
3	2	elrab3 1906	. . 3 |- (A e. B -> (A e. {x e. B | x (_ A} <-> A (_ A))
4	1, 3	mpbiri 194	. 2 |- (A e. B -> A e. {x e. B | x (_ A})
5		sseq1 2082	. . . . . 6 |- (x = y -> (x (_ A <-> y (_ A))
6	5	elrab 1905	. . . . 5 |- (y e. {x e. B | x (_ A} <-> (y e. B /\ y (_ A))
7	6	pm3.27bi 326	. . . 4 |- (y e. {x e. B | x (_ A} -> y (_ A)
8	7	rgen 1698	. . 3 |- A.y e. {x e. B | x (_ A}y (_ A
9		ssunieq 2531	. . . 4 |- ((A e. {x e. B | x (_ A} /\ A.y e. {x e. B | x (_ A}y (_ A) -> A = U.{x e. B | x (_ A})
10	9	eqcomd 1480	. . 3 |- ((A e. {x e. B | x (_ A} /\ A.y e. {x e. B | x (_ A}y (_ A) -> U.{x e. B | x (_ A} = A)
11	8, 10	mpan2 696	. 2 |- (A e. {x e. B | x (_ A} -> U.{x e. B | x (_ A} = A)
12	4, 11	syl 10	1 |- (A e. B -> U.{x e. B | x (_ A} = A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648   (_ wss 2047  U.cuni 2503

This theorem is referenced by:  fctopOLD 7650  cctop 7652  isopn3 7697  chsupid 9311  shatomistic 10288

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rab 1652  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504

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